10.2.3: The Connection Between the Stream Function and the Potential Function
- Page ID
- 779
For this discussion, the situation of two dimensional incompressible is assumed. It was shown that
\[
\label{if:eq:UxpotentianlSteam}
\pmb{U}_x = \dfrac{\partial \phi}{\partial x}
= \dfrac{\partial \psi}{\partial y}
\]
and
\[
\label{if:eq:UypotentianlSteam}
\pmb{U}_y = \dfrac{\partial \phi}{\partial y}
= - \dfrac{\partial \psi}{\partial x}
\]
These equations (70) and (71) are referred to Definition of the potential function is based on the gradient operator as \(\pmb{U} = \boldsymbol{\nabla}\phi\) thus derivative in arbitrary direction can be written as
\[
\label{if:eq:arbitraryPotential}
\dfrac{d\phi}{ds} = \boldsymbol{\nabla}\phi \boldsymbol{\cdot} \widehat{s} =
\pmb{U} \boldsymbol{\cdot} \widehat{s}
\]
\[
\label{if:eq:velocitystreamPotential}
{U} = \dfrac{d\phi}{ds}
\]
If the derivative of the stream function is chosen in the direction of the flow then as in was shown in equation (58). It summarized as
\[
\label{if:eq:streamFpotentialFDerivative}
\dfrac{d\phi}{ds} = \dfrac{d\psi}{dn}
\]
Example 10.3
Solution 10.3
\[
\label{streamTOpotential:derivativeY}
\dfrac{\partial \phi}{\partial x} =
\dfrac{\partial \psi}{\partial y} = - 2\, y
\]
\[
\label{streamTOpotential:integralX}
\phi = - 2\,x\,y + f(y)
\]
where \(f(y)\) is arbitrary function of \(y\). Utilizing the other relationship (??) leads
\[
\label{streamTOpotential:eq:derivativeX}
\dfrac{\partial \phi}{\partial y} = - 2\, x + \dfrac{d\,f(y) }{dy} =
- \dfrac{\partial \psi}{\partial x} = - 4\,x^3
\]
Therefore
\[
\label{streamTOpotential:eq:potentialODE}
\dfrac{d\,f(y) }{dy} = 2\,x - 4\, x^3
\]
After the integration the function \(\phi\) is
\[
\label{streamTOpotential:phiIntegration}
\phi = \left( 2\,x - 4\, x^3 \right)\, y + c
\]
The results are shown in Figure
Contributors and Attributions
Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.